Solvmanifolds and noncommutative tori with real multiplication (Q959505)

Let a real quadratic field be given. \textit{M. F. Atiyah, H. Donnelly} and \textit{I. M. Singer} [Ann. Math. II. Ser. 118, 131--177 (1983; Zbl 0531.58048)] related the \(\eta\)-invariant of the signature operator on a particular three-dimensional solvmanifold \(X\) to the Shimizu \(L\)-function of the field, thus proving a conjecture of Hirzebruch. In the present paper the author considers the spectral triple defined by the signature operator on the corresponding noncommutative torus with real multiplication. As in the classical case the \(\eta\)-invariant of the triple vanishes. The main result of the paper is that the \(\eta\)-invariant of a restriction of the operator can be expressed in terms of the Shimizu \(L\)-function. Its proof is based on methods of loc. cit. The relation between both settings is described in the paper as follows: The solvmanifold \(X\) is a torus bundle over \(S^1\). Using the isospectral deformation formalism of \textit{A. Connes} and \textit{G. Landi} [Commun. Math. Phys. 221, No. 1, 141--159 (2001; Zbl 0997.81045)] its algebra of functions can be deformed in direction of the fibers. The vertical signature operator on the deformed fibers yields the spectral triple on the noncommutative torus. The author also gives a different interpretation of the Shimizu \(L\)-function: She introduces Lorentzian spectral triples over real quadratic fields and their \(\eta\)-invariants. Then she defines a Lorentzian spectral triple associated with a twisted group algebra of the fundamental group \(\pi_1(X)\) and relates its \(\eta\)-invariant to the Shimizu \(L\)-function. On the way, by using the Baum-Connes assembly map and a twisted index theorem, the author calculates the range of the canonical trace from \(K_0(C^*(\pi_1(X),\sigma))\) to \(\mathbb{R}\) for group cocycles \(\sigma\) of a particular form. Here \(C^*(\pi_1(X),\sigma)\) is the twisted group \(C^*\)-algebra. This result is of interest in connection with gap labeling problems.